Permutations as Angular Data: Efficient Inference in Factorial Spaces

Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities (n!). It makes the direct definition of a multinomial distribution over permutation space impractical for all but a very small n. In this work we propose an embedding of all n! permutations for a given n in a surface of a hyper sphere defined in ℝ^(n-1). As a result of the embedding, we acquire ability to define continuous distributions over a hyper sphere with all the benefits of directional statistics. We provide polynomial time projections between the continuous hyper sphere representation and the n!-element permutation space. The framework provides a way to use continuous directional probability densities and the methods developed thereof for establishing densities over permutations. As a demonstration of the benefits of the framework we derive an inference procedure for a state-space model over permutations. We demonstrate the approach with simulations on a large number of objects hardly manageable by the state of the art inference methods, and an application to a real flight traffic control dataset.